Assumption Of Small Deformations Research Articles

Rayleigh-Wave Dispersion Analysis and Inversion Based on the Rotation.

Rotational observation is essential for a comprehensive description of the ground motion, and can provide additional wave-field information. With respect to the three typical layered models in shallow engineering geology, under the assumption of linear small deformation, we simulate the 2-dimensional radial, vertical, and rotational components of the wave fields and analyze the different characteristics of Rayleigh wave dispersion recorded for the rotational and translational components. Then, we compare the results of single-component inversion with the results of multi-component joint inversion. It is found that the rotational component has wider spectral bands and more higher modes than the translational components, especially at high frequencies; the rotational component has better anti-interference performance in the noisy data test, and it can improve the inversion accuracy of the shallow shear-wave velocity. The field examples also show the significant advantages of the joint utility of the translational and rotational components, especially when a low-velocity layer exists. Rotational observation shall be beneficial for shallow surface-wave exploration.

Simulation and Experiment Study on the Bridge Deflection Measuring Method Based on Secant Inclination

Deflection is the most direct indicator that reflects the bearing capacity of the bridge and the overall stiffness. There are many ways to measure the deflection of Bridges, and the inclination angle method is the most commonly used indirect method, but the existing theory of inclination angle method is relatively complicated. Based on the facts of the bridge small inclination, this article proposes the method of obtaining the bridge deflection by the inclination of the secant line constructed from the adjacent measurement points. Firstly, according to the bending deformation curve of general simply supported beam, the deflection calculation formula of each measuring point is derived based on the assumption of small deformation and the inclination Angle of measuring point. Secondly, a large commercial finite element software ANSYS 10.0 is used to carry out numerical simulation on the simply-supported beam under concentrated load in mid-span, and the deflection results of the numerical simulation are compared and verified with the theoretical results of the proposed method. Finally, the measured deflection results of the simply-supported beam model under mid-span load are compared with the theoretical results of the proposed method. The verification results show that if the actual model is consistent with the theoretical model, the proposed method has good accuracy.

ANSYS implementation of a coupled 3D peridynamic and finite element analysis for crack propagation under quasi-static loading

This study presents an approach to couple peridynamic (PD) and finite element method (FEM) for 3D deformation and crack propagation in ANSYS framework. The stiffness matrix of the PD element is derived by using the peridynamic least squares minimization (PD LSM) method based on the small deformation assumption. The PD governing equations are constructed by using MATRIX 27 element native to ANSYS. The coupling between MATRIX 27 elements and traditional finite elements is achieved through the coupled degrees of freedom (DOF) command available in ANSYS. The crack propagation is implemented by monitoring the number of broken bonds and updating the stiffness matrix of the PD element. The accuracy of the coupled PD-FE approach is verified by comparison against the 3D FE displacement predictions. The validity of the coupled PD-FE approach is established by simulating the wedge splitting test.

Study on Ground Deformation During Shield Tunnel Construction

Through the systematic analysis of the ground settlement generatedby the process of shield tunneling, the relationships between grounddeformation and construction parameters are studied in this paper. Basedon the assumption of linear small deformation, a mathematical model ofthe relationship between ground deformation and construction parametersis set up. The principle and method of optimization for estimating grounddeformation is studied. The actual measured data are compared with theresults of theoretical analysis in a case. Considering different groundformations in different construction sites with different adverse effectson surface and underground structures, the ground surface deformationscaused by shield tunneling is an aimed topic in this paper. The contributionsand research implications are the revealed relationships between the grounddeformation and the shield tunneling parameters during construction.

Modified Stoney formula for determining stress within thin films on large-deformation isotropic circular plates

The Stoney formula is widely used to obtain residual stress in films on isotropic circular plates. However, in the case of large deformations, this formula produces significant errors because of the assumption of small deformations in its derivation. In this study, a modified Stoney formula that extends its scope of application to the nonlinear domain is proposed. A one-phase exponential decay function with a single coefficient p is used to relate the curvature of the substrate to the stress in the film. The coefficient p can be expressed as a function of the thickness, the diameter, Young’s modulus, and Poisson’s ratio of the circular plate. A linear fitting technique is applied to ascertain the relationship between the coefficient p and the aforementioned parameters. The modified Stoney formula is simple, yet accurate, and can be used to calculate the residual stress in the film directly from the measured curvature of isotropic circular plates with various dimensions and materials.

Coupled vortex-induced modeling for spatially large-curved beam with elastic support

The existing studies on mathematical modeling for the forced vibration response of naturally curved beams under elastic boundary conditions mainly focus on slightly curved structures. This work proposes the mathematical model for naturally curved beam with a large curvature under the elastic boundary. Under the assumption of small deformations, the six degrees of freedom are reduced to four degrees of freedom through spatial bending beam theory and differential geometry. The dynamics governing equations of the naturally curved beam are derived using Hamilton’s principle and Frenet–Serret formula. An improved Van der Pol model is introduced to investigate the vortex induced vibration (VIV) or vortex induced divergence (VID) response. Hammer test, wind tunnel experiment and finite element method validate the proposed model for spatially curved helical beam. It is found that under the fixed boundary condition, transposition occurs in the first two coupled modes in the normal direction of the helical beam. The errors of the first two nature frequencies are 0.016% and 0.088%, respectively. The modal analysis under the elastic boundary shows that the first-order decoupled modes in each direction will “disappear” with the variation of stiffness. Under the boundary condition of small stiffness, the first two modes of the normal direction change abruptly at the subtended angle of 76∘∼90∘. This abrupt change produces negative damping and leads to structural unstability. When the first two frequencies are close to each other, the water velocity region of VIV or VID response will intersect and the orthogonal modes in normal direction should be calculated coupled. In this case, the amplitude of the first order mode in the normal direction will always be greater than that of the second order mode. The maximum amplitude in u direction can reach 0.014 m. The obtained results provide useful information for further research on the fluid-induced vibration response of natural large-curved beam .

Theoretical analysis and optimization of toggle-brace damper for cable tray system

The cable tray system, one type of non-structural components, may suffer severe damage and even fall in case of earthquakes, causing interruptions to post-earthquake operations and even injuries or casualties. This paper introduces a toggle-brace damper (TBD) into the cable tray system to control its seismic response and damage. However, owing to the large deformation/rotation effect in the TBD equipped cable tray system, the traditional design method based on small deformation assumption is no longer applicable. The new deformation calculations of suspension frame and TBD subjected to displacement loading are established combining the optimization technique and the geometry relationship, respectively. After then, an optimization design method for connecting location of the damper brace is proposed, considering the installation errors, deformation constraints, and loading process. With the aid of numerical analysis, it is found that the large deformation effect and in-plane installation errors have considerable influence on the damping magnification function. Compared with typical seismic resistant elements such as the steel brace, diagonal- and chevron-brace damper, the proposed optimal TBD can dissipate more energy and effectively suppress the vibration of cable tray system.

A novel nonlinear finite element method for structural dynamic modeling of spacecraft under large deformation

An accurate structural dynamic model is significant for control system design and safety of the spacecraft. Traditional linear structural dynamic models mostly based on the hybrid coordinate method, requiring the existence of central rigid bodies and the satisfaction of the small deformation assumption. However, modern spacecraft are utilizing more and more flexible structures and payloads, leading to a possibility of non-rigid-body tendency and large deformation, which needs a nonlinear solution. Existing nonlinear finite element methods become difficult to implement structural analysis for spacecraft due to the high degrees of freedom of the elements. Therefore, this paper proposes an accurate, freedom-reduced, universally applicable, and full-flexible nonlinear finite element method for modern spacecraft based on the continuum mechanics, taking the triangular element as an example. Simulations verify that this method can effectively solve the structural motion of large deformation, and is able to calculate the time-domain response under typical loads such as concentrated forces, body forces and torques.

Immersed boundary thin shell analysis using 3D B-Spline background mesh

An immersed boundary shell element formulation based on Kirchhoff-Love shell theory is presented here that uses a background mesh consisting of uniform 3D B-spline elements within which the geometry and the boundaries are embedded. In this theory, rotation is assumed to be identical to slope which introduces second derivatives of deflection in the weak form. As a result, elements that provide tangent continuous approximation of the displacement are needed. A 3D shell element is formulated under small deformation assumptions that has three translational degrees of freedom (DOF) per node and uses quadratic B-Spline shape functions. The displacement field is transformed to a local coordinate system attached to the shell surface and defined such that the shell theory assumptions are satisfied. Essential boundary conditions are applied using the step boundary method where approximate step functions are utilized to construct trial and test solutions that satisfy the boundary conditions. Benchmark problems that are commonly used to test shell elements are used to validate the 3D B-spline thin shell element presented here. • 3D B-spline element formulation for modeling thin shells using an immersed boundary approach. • The shell geometry is immersed in a background mesh. • The background mesh consists of isoparametric B-spline elements. • The nodes of the element have three translational degrees of freedom and no rotations. • Step boundary method is used for applying boundary conditions.

Elastoplastic topology optimization of cyclically loaded structures via direct methods for shakedown

For the first time, the lower bound shakedown theorem is integrated into a level set–based topology optimization framework to identify lightweight elastoplastic designs. Shakedown is a cyclic elastoplastic behavior in which, upon cycling beyond the elastic limit, the accumulation of plastic strain arrests and purely elastic behavior is recovered. In contrast to most elastoplastic topology optimization, the use of a lower bound shakedown limit allows elastoplastic shakedown limits to be rigorously estimated using only the elastic solution. Under small deformation assumptions, this amounts to solving one simple partial differential equation, avoiding the non-linearity associated with plasticity, and thus simplifying the resolution process. Numerical results are provided for several benchmark examples. The results highlight the design performance enhancements attributed to allowing elastoplastic shakedown to occur instead of designing to first yield. In particular, up to 10% reduction in weight is found for the simple structures considered.

Dynamics of 3D sliding beams undergoing large overall motions

This paper presents the 3D dynamic formulations for a flexible beam sliding through a revolute-prismatic joint. Considering the geometric nonlinearity, the configuration space of the 3D flexible beam is a nonlinear differentiable manifold (R3×SO(3)). Moreover, the beam manipulated by the revolute-prismatic joint can undergo large overall motion and slide through the joint. Because of the difficulty mentioned above, most studies on these problems focus on 2D cases or are tackled under a small deformation assumption. In this paper, the rotation matrices are parameterized using rotational vectors to describe accurately the spatial configuration of flexible beams. For convenience, to describe the finite deformation of the beams, the material frame is fixed on the revolute-prismatic joint but will change over time. The corotational method is introduced to take the geometric nonlinearity (small strain and large rotation) of the beam into account. In the corotational frame, the strain energy and kinetic energy of the elements are derived with the same shape functions, which are used to describe the local displacements, to maintain the element-independent framework. Then a ‘standard element’ can be embedded within this framework. In order to consider the shear deformation, the flexible beam is discretized using a fixed number of variable-domain interdependent interpolation elements. Rotary inertia is also considered in this paper. The nonlinear equations of motion are derived by using the extended Hamilton's principle and solved by using the Hilber-Hughes-Taylor method and the Newton-Raphson iteration method. Four examples are presented to demonstrate the validity, accuracy and versatility of the present dynamic formulation.

Bond-Based Peridynamics with Stretch and Rotation Kinematics for Opening and Shearing Modes of Fracture

This study presents a new bond-based peridynamic approach for modeling the elastic deformation of isotropic materials with bond stretch and rotation, thus removing the constraint on the Poisson’s ratio. The resulting PD equilibrium equations derived under the assumption of small deformation are solved by employing implicit techniques. The bond constants associated with stretch and rotation kinematic are directly related to the constitutive relations of stress and strain components in continuum mechanics. Also, the expressions for the critical stretch and critical relative rotation are derived in terms of mode I and mode II critical energy release rates, respectively. Lastly, it does not require a surface correction procedure, and the displacement and traction type boundary conditions are directly imposed without introducing fictitious regions in the domain. The capability of this approach is first demonstrated by capturing the correct deformation of plate type structures under general loading conditions. Subsequently, its capability for failure prediction is established by simulating the response of a double cantilever beam (DCB) under mode I type loading and compact shear specimen under mode II type loading.

Modeling of the crystallization process of a polymer plate taking into account large deformations

The issues of modeling the thermomechanical behavior of a polymer material during cooling accompanied by crystallization are considered. The proposed approach is applied to the problem of deformation upon cooling of an infinite plate made of low-pressure polyethylene during crystallization. Mathematical formulations of thermal kinetic and thermomechanical boundary value problems are presented. The results of the numerical solution of a coupled heat-kinetic problem, including the non-stationary heat conduction equation and the crystallization kinetics equation obtained under the assumption that the characteristics of the material depend on temperature are given. When solving the boundary value problem of thermomechanics, the previously obtained phenomenological nonlinear constitutive relations are used, which continuously describe the behavior of a polymer material in a wide temperature range, including the range of phase transformations. The construction of constitutive relations was carried out using the Peng-Landel potential. A weak variational formulation of the boundary value problem, constructed using the Galerkin method, is presented. A linearization procedure is carried out, which makes it possible to reduce the solution of an initially nonlinear boundary value problem to solving a sequence of linear problems with relatively small increments of the displacement vector components. In this case, the linearization of geometric relations is performed by superimposing small increments of deformations on finite ones, linearization of the constitutive relations - expansion in a Taylor series with subsequent retention of linear terms under the assumption that the increment of the components of the strain tensor is small. An approach is demonstrated that naturally takes into account the small increments of temperature and structural deformations arising in the material. Some aspects of the numerical implementation of the created algorithm based on the finite element technology of constructing a discrete analogue of the problem under consideration are analyzed. The results of solving the problem under discussion in a linear formulation under the assumption of small deformations and in a nonlinear formulation are compared.

Modal reduction procedures for flexible multibody dynamics

Through a critical review of the various component mode synthesis techniques developed in the past, it is shown that both Craig–Bampton’s and Herting’s methods are particular cases of the mode-acceleration method and furthermore, Rubin’s method is equivalent to Herting’s method. Consequently, the mode-acceleration method is the approach of choice due to its simplicity and because unlike the other methods, it imposes no restriction on the selection of the modes. Next, a general approach to the modal reduction of geometrically nonlinear structures is developed within the framework of the motion formalism, based on the small deformation assumption. The floating frame of reference is defined unequivocally by imposing six linear constraints on the deformation measures, which are defined as the vectorial parameterization of the relative motion tensor that brings the fictitious rigid-body configuration to its deformed counterpart. This approach yields deformation measures that are both objective and tensorial, unlike their classical counterparts that share the first property only. Derivatives are expressed in the material frame, leading to computationally advantageous properties: tangent matrices are functions of the deformation measures only and become nearly constant during the simulation. Numerical examples demonstrate the accuracy, robustness, and numerical efficiency of the proposed approach. With a small number of modal elements, the formulation is able to capture geometrically nonlinear effects accurately, even in the presence of inherently nonlinear phenomena such as buckling.

Run-Out Simulation of a Landslide Triggered by an Increase in the Groundwater Level Using the Material Point Method

Deformation mechanisms of the slopes are commonly schematized in four different stages: pre-failure, failure, post-failure and eventual reactivation. Traditional numerical methods, such as the finite element method and the finite difference method, are commonly employed to analyse the slope response in the pre-failure and failure stages under the assumption of small deformations. On the other hand, these methods are generally unsuitable for simulating the post-failure behaviour due to the occurrence of large deformations that often characterize this stage. The material point method (MPM) is one of the available numerical techniques capable of overcoming this limitation. In this paper, MPM is employed to analyse the post-failure stage of a landslide that occurred at Cook Lake (WY, USA) in 1997, after a long rainy period. Accuracy of the method is assessed by comparing the final geometry of the displaced material detected just after the event, to that provided by the numerical simulation. A satisfactory agreement is obtained between prediction and observation when an increase in the groundwater level due to rainfall is accounted for in the analysis.

Port-Hamiltonian flexible multibody dynamics

A new formulation for the modular construction of flexible multibody systems is presented. By rearranging the equations for a flexible floating body and introducing the appropriate canonical momenta, the model is recast into a coupled system of ordinary and partial differential equations in port-Hamiltonian (pH) form. This approach relies on a floating frame description and remains valid under the assumption of small deformations. This allows including mechanical models that cannot be easily formulated in terms of differential forms. Once a pH model is established, a finite element based method is then introduced to discretize the dynamics in a structure-preserving manner. Thanks to the features of the pH framework, complex multibody systems are constructed in a modular way. Constraints are imposed at the velocity level, leading to an index 2 quasi-linear differential-algebraic system. Numerical tests are carried out to assess the validity of the proposed approach.

Damage-Plastic Constitutive Model of Thin-Walled Circular Steel Tubes for Space Structures

AbstractIn this paper, we establish a coupled damage-plastic constitutive model in the scheme of small deformation assumption, based on a continuum damage mechanics model proposed by Lemaitre, for ...

Dynamic response determination of viscoelastic annular plates using FSDT – perturbation approach

In this paper, the transient response of a viscoelastic annular plate which has time-dependent properties is determined mathematically under dynamic transverse load. The axisymmetric conditions are considered in the problem. The viscoelastic properties obey the standard linear solid model in shear and the bulk behavior in elastic. The equations of motion are extracted using Hamilton’s principle by small deformation assumption for the elastic condition and they are extended to the viscoelastic form by defining viscoelastic operators based on the separating the bulk and shear behaviors. The displacement field is defined with the first order shear deformation theory by considering the transverse normal strain effect. These equations which contain four coupled partial differential equations with variable coefficients are solved using the perturbation technique. The results are compared with those obtained from the classical plate theory and the finite element method. The presented formulation is useful for parametric study because it does not need to generate mesh and selecting time step for each model; also the running time is short with respect to the finite elements method. For sensitivity analysis, the effects of geometrical and mechanical parameters on the response are investigated by parametric study.

Geometrical Nonlinear Effects in the Dynamic Response of Sandwich Beams

AbstractThe geometrical nonlinear dynamic response of sandwich beams is studied using a dynamic high-order nonlinear model. The model is derived using the variational principle of virtual work and uses the Extended High-Order Sandwich Panel Theory approach with consideration of two interfaces between the three layers. A first-order shear deformation theory is adopted for the face sheets, while the kinematic assumption of high-order small deformations that account for out-of-plane compressibility are considered for the core layer. The nonlinearity of the dynamic model is introduced by considering geometrically nonlinear kinematic relations in the face sheets. The nonlinear kinematic relations and the dynamic modeling aim to evaluate the effects of the two features and their coupling on the response. The nonlinear dynamic response of sandwich beams is studied through two numerical cases and comparison of the nonlinear results with their linear counterparts. The first case looks into the coupling of the global geometrical nonlinear behavior with the dynamic behavior. The second case focuses on the local instability of the face sheets and its interaction with the compressibility of the core in the dynamic response of soft core sandwich beams. The comparison of linear and nonlinear dynamic response in the two cases sheds light on the coupling of the geometrical nonlinear and dynamic behaviors. Among other features, the latter is expressed by nonlinear attractors, higher modes response, nonlinear frequency response, and significant wrinkling response.

Structures and mechanical behaviors of soft nanotubes confining adhesive single or multiple elastic nanoparticles

The mechanical interaction between soft membrane nanotubes and confined nanoparticles plays essential roles in numerous cellular activities such as inter- and intracellular transport and signaling, membrane tubulation and pearling transformation driven by adsorbed nanocomponents, and cooperative cell uptake. The aim of this work is to theoretically analyze how structures and mechanical behaviors of particle-in-tube complexes are regulated by the particle elasticity, covering the cases of a single particle and multiple particles of spatial periodicity. Depending on the interplay of membrane tension, adhesion energy, nanoparticle size, and the rigidity ratio between the nanoparticles and soft tubes, characteristic interaction states in terms of wrapping degrees of the confined particles have been identified, and the corresponding wrapping phase diagrams are determined. As the particles become softer, the adhesion energy required for the partial-wrapping state becomes lower but that for full-wrapping is larger, and a rich variety of characteristic system configurations are observed including the bulged or undulated cylinders, necklaces of spheres or prolates, and sausage-string shapes. Moreover, two fundamental modes of interaction between multiple nanoparticles and the soft nanotube are revealed, the cooperative wrapping and individual wrapping. Based on perturbation analysis at small tube deformation and configurational assumptions of catenoid, torus, and one-sheeted hyperboloid at more general tube deformation, we have also obtained analytical solutions on the wrapping of rigid spherical nanoparticles in soft tubes. Comparison between wrapping of elastic particles by a flat membrane and by a soft membrane tube is discussed. This study enriches our knowledge on the diversity of membrane tubules in morphologies and structures.